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Properties

Label 2-819-117.2-c1-0-11
Degree $2$
Conductor $819$
Sign $0.207 - 0.978i$
Analytic cond. $6.53974$
Root an. cond. $2.55729$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.63 − 1.63i)2-s + (1.73 + 0.0399i)3-s + 3.37i·4-s + (−1.02 + 3.84i)5-s + (−2.77 − 2.90i)6-s + (0.258 − 0.965i)7-s + (2.25 − 2.25i)8-s + (2.99 + 0.138i)9-s + (7.98 − 4.60i)10-s + (−0.994 + 0.994i)11-s + (−0.135 + 5.84i)12-s + (−2.45 + 2.64i)13-s + (−2.00 + 1.15i)14-s + (−1.93 + 6.60i)15-s − 0.647·16-s + (−1.81 + 3.14i)17-s + ⋯
L(s)  = 1  + (−1.15 − 1.15i)2-s + (0.999 + 0.0230i)3-s + 1.68i·4-s + (−0.460 + 1.71i)5-s + (−1.13 − 1.18i)6-s + (0.0978 − 0.365i)7-s + (0.797 − 0.797i)8-s + (0.998 + 0.0461i)9-s + (2.52 − 1.45i)10-s + (−0.299 + 0.299i)11-s + (−0.0389 + 1.68i)12-s + (−0.680 + 0.732i)13-s + (−0.536 + 0.309i)14-s + (−0.499 + 1.70i)15-s − 0.161·16-s + (−0.440 + 0.762i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.207 - 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.207 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.207 - 0.978i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (470, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.207 - 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.530811 + 0.429842i\)
\(L(\frac12)\) \(\approx\) \(0.530811 + 0.429842i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.73 - 0.0399i)T \)
7 \( 1 + (-0.258 + 0.965i)T \)
13 \( 1 + (2.45 - 2.64i)T \)
good2 \( 1 + (1.63 + 1.63i)T + 2iT^{2} \)
5 \( 1 + (1.02 - 3.84i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (0.994 - 0.994i)T - 11iT^{2} \)
17 \( 1 + (1.81 - 3.14i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.52 + 5.70i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-1.33 + 2.31i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 7.46iT - 29T^{2} \)
31 \( 1 + (4.03 + 1.08i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (1.05 - 3.94i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (9.93 - 2.66i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (3.32 - 1.91i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.64 - 6.12i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 - 3.56iT - 53T^{2} \)
59 \( 1 + (-3.49 + 3.49i)T - 59iT^{2} \)
61 \( 1 + (-2.96 - 5.13i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.38 - 5.16i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-1.51 + 0.404i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (-0.403 - 0.403i)T + 73iT^{2} \)
79 \( 1 + (-3.53 + 6.12i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (12.1 - 3.26i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (1.61 + 0.433i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-17.1 - 4.59i)T + (84.0 + 48.5i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.43320988018373394566790716991, −9.742958076662130919004432232156, −8.864053197030824574585594781048, −8.104060935998309263191218612240, −7.12964725761666107134777967351, −6.82353452354054452031530134081, −4.48410564558529515518746356760, −3.41083626346326813922168591634, −2.71381369916343822871825530571, −1.86867428844564072353244562853, 0.42262689355623123640638772921, 1.87003746457223401683290633792, 3.67991939018094955091505599918, 4.97908296594641685716952780177, 5.65004345613723987134600880089, 7.06257649021402092297678364082, 7.892997204533136080774301989778, 8.302229157850846443650016753887, 8.907100549955144462753630084362, 9.598189294108898875073352443346

Graph of the $Z$-function along the critical line